A maximum linear matroid parity set is called a basic matroid parity
set, if its size is the rank of the matroid. We show that
determining the existence of a common base (basic matroid parity
set) for linear matroid intersection (linear matroid parity) is in
NC^2, provided that there are polynomial number of common bases
(basic matroid parity sets). For graphic matroids, we show that
finding a common base for matroid intersection is in NC^2, if the
number of common bases is polynomial bounded. To our knowledge,
these algorithms are the first deterministic NC algorithms for
matroid intersection and matroid parity. We also give a new RNC^2
algorithm that finds a common base for graphic matroid intersection.

Similar to the Tutte's theorem, we derive the determinant criterion
for the existence of a common base (basic matroid parity set) for
linear matroid intersection (linear matroid parity). Moreover, we
prove that if there is a black-box NC algorithm for PIT
(Polynomial Identity Testing), then there is an NC algorithm to
determine the existence of a common base (basic matroid parity set)
for linear matroid intersection (linear matroid parity).

Let two linear matroids have the same rank in matroid intersection.
A maximum linear matroid intersection (maximum linear matroid parity
set) is called a basic matroid intersection (basic matroid parity
set), if its size is the rank of the matroid. We present that
enumerating all basic matroid intersections (basic matroid parity
sets) is in NC^2, provided that there are polynomial bounded basic
matroid intersections (basic matroid parity sets). For the graphic
matroids, We show that constructing all basic matroid intersections
is in NC^2 if the number of basic graphic matroid intersections is
polynomial bounded. To our knowledge, these algorithms are the first
deterministic NC-algorithms for matroid intersection and matroid
parity. Our result also answers a question of Harvey \cite{HAR}.